A282472 -id:A282472 - cp's OEIS frontend

A346016 Numbers k where the average number of distinct prime factors of the divisors of k sets a new record.

1, 2, 4, 6, 12, 24, 30, 60, 120, 180, 210, 420, 840, 1260, 2310, 4620, 9240, 13860, 27720, 30030, 60060, 120120, 180180, 360360, 510510, 1021020, 2042040, 3063060, 6126120, 9699690, 19399380, 38798760, 58198140, 116396280, 223092870, 446185740, 892371480, 1338557220

Offset: 1

Amiram Eldar, Jul 01 2021

nonn

First differs from A282472 at n = 48.
The corresponding record values are 0, 1/2, 2/3, 1, 7/6, 5/4, 3/2, 5/3, 7/4, 11/6, 2, ...
All the terms are least integers of their prime signature (A025487) since f(n) = A346009(n)/A346010(n) depends only on the prime signature of n.
This sequence is infinite since f(n) is unbounded. E.g., f(A002110(n)) = n/2.

Amiram Eldar, Table of n, a(n) for n = 1..198

Cf. A002110, A282472, A346009, A346010.

Subsequence of A025487.

lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]; f[p_, e_] := e/(e + 1); f[1] = 0; f[n_] := Plus @@ f @@@ FactorInteger[n]; s = {}; fmax = -1; Do[f1 = f[lps[[k]]]; If[f1 > fmax, fmax = f1; AppendTo[s, lps[[k]]]], {k, 1, Length[lps]}]; s
DeleteDuplicates[Table[{n,Mean[PrimeNu[Divisors[n]]]},{n,100000}],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]] (* The program generates the first 21 terms of the sequence. *) (* Harvey P. Dale, Nov 24 2024 *)

A368252 Numbers with a record value of number of uniform divisors (A327527).

Original entry on oeis.org

Offset: 1

Amiram Eldar, Dec 19 2023

nonn

First differs from A346016 at n = 48, and from A282472 at n = 155.

			The sequence A327527 starts with 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 5, ... . The record values, 1, 2, 3, 4 and 5, occur at indices 1, 2, 4, 6, 12, the first 5 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..212
Amiram Eldar, Table of n, a(n), A327527(a(n)) for n = 1..212

Cf. A282472, A327527, A346016.

Subsequence of A025487.

d[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, 1 + Total[2^Accumulate[Count[e, #] & /@ Range[Max[e], 1, -1]] - 1]]; d[1] = 1;
v = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]];
seq = {}; dm = 0; Do[If[(dk = d[v[[k]]]) > dm, dm = dk; AppendTo[seq, v[[k]]]], {k, 1, Length[v]}]; seq

cp's OEIS Frontend

A346016 Numbers k where the average number of distinct prime factors of the divisors of k sets a new record.

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